Using the Formulas

What this move does

Read the multiplication formulas backwards, and they become factoring formulas:

a^{2} - b^{2} a^{2} \pm 2 ab + b^{2} a^{3} \pm b^{3} = (a + b) (a - b) = (a \pm b)^{2} = (a \pm b) (a^{2} \mp ab + b^{2})

The workbench above shows the first one: cut a corner of side $b$ out of a square of side $a$ . Slice the L-shape that is left, turn one piece around, and you get exactly a rectangle of length $a + b$ and width $a - b$ . The area has not changed, so $a^{2} - b^{2} = (a + b) (a - b)$ .

Count the terms: two terms suggest a difference of squares, or a sum or difference of cubes; three terms suggest a perfect square.
Identify the parts: what is $a$ and what is $b$ ? $4 x^{2}$ is $(2 x)^{2}$ , and $27$ is $3^{3}$ .
Use the formula, then check whether each factor can be split further.

Predict first

Without writing anything down, work out $5 1^{2} - 4 9^{2}$ in your head.

Choose an answer first; then you can see the result.

Work through one: difference of squares

Example: factor x² − 9

Step 1 / 5

Spot two squares
$x^{2}$ is the square of $x$ , $9$ is the square of $3$ , and there is a minus sign between them. In the picture, $a$ is the side of the big square and $b$ is the side of the small one.

$x^{2} - 3^{2}$
Cut out a corner
Slice it
Turn it and attach it
Back to the problem

Perfect squares

With three terms, first check whether the first and last terms are squares, then whether the middle term is twice their product.

Four pieces: one

a^{2}

, two

ab

and one

b^{2}

. Click “Make a square”. Then click “Remove the two ab” and see how much is missing.

A negative middle term works the same way; you just read the picture differently. Take two strips $ab$ away from $a^{2}$ : the small corner at the bottom right is taken away twice, so one $b^{2}$ has to be put back.

(a - b)^{2} = a^{2} - 2 ab + b^{2}

. Click 1 → 4 in turn.

Difference and sum of cubes

A square is an area; a cube is a volume. Cut a small cube out of a big cube, and what is left can be taken apart into three slabs of the same thickness.

a³ − b³ as three slabs

Step 1 / 3

Cut out a corner
From a cube of edge $a$ , cut out a corner cube of edge $b$ (the red dashed lines). The volume left is $a^{3} - b^{3}$ . Use “Turn it” to look from another angle.

$a^{3} - b^{3}$
Cut it in three
Line them up

You can see the sum of cubes too. Stack the small cube on the big one, then pull a slab off the big cube and move it on top: you get a column of height $a + b$ .

a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})

. Click 1 → 2 → 3. Why is the base area of the column

a^{2} - ab + b^{2}

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: The difference of squares formula factors it

$a^{2} - b^{2}$ : the L-shape can always be rearranged into a rectangle.

Practice

Exercise 1Basic

Factor $4 x^{2} - 25$ .

Exercise 2Basic

Factor $x^{2} - 12 x + 36$ .

Exercise 3Advanced

Factor $x^{3} + 27$ .

Exercise 4Challenge

Factor $x^{6} - 1$ completely. (You should get four factors.)

What this move does

Work through one: difference of squares

Example: factor x² − 9

Perfect squares

Difference and sum of cubes

a³ − b³ as three slabs

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

The square of a sum = the sum of the squares?

Using a formula once and stopping

The signs in the difference of cubes